View source for Physics:Lee–Yang theorem

In [[Physics:Statistical mechanics|statistical mechanics]], the '''Lee–Yang theorem''' states that if [[Physics:Partition function (statistical mechanics)|partition functions]] of certain  models in [[Physics:Statistical field theory|statistical field theory]] with ferromagnetic interactions are considered as functions of an external field, then all zeros 
are  purely imaginary (or on the unit circle after a change of variable). The first version was proved for the [[Physics:Ising model|Ising model]] by {{harvs|txt|first1=T. D.| last1=Lee|first2=C. N. |last2=Yang|year=1952}} {{harv|Lee|Yang|1952}}. Their result was later extended to more general models by several people. Asano in 1970 extended the Lee–Yang theorem to the [[Physics:Heisenberg model (quantum)|Heisenberg model]] and provided a simpler proof using [[Asano contraction]]s. {{harvtxt|Simon|Griffiths|1973}} extended the Lee–Yang theorem to certain continuous probability distributions by approximating them by a superposition of Ising models. {{harvtxt|Newman|1974}} gave a general theorem stating roughly that the Lee–Yang theorem holds for a ferromagnetic interaction provided it holds for zero interaction. {{harvtxt|Lieb|Sokal|1981}} generalized [[Biography:Charles M. Newman|Newman]]'s result from measures on '''R''' to measures on higher-dimensional Euclidean space.

There has been some speculation about a relationship between the Lee–Yang theorem and the [[Riemann hypothesis]] about the [[Riemann zeta function]]; see {{harv|Knauf|1999}}.

==Statement==

===Preliminaries===
Along the formalization in {{harvtxt|Newman|1974}} the Hamiltonian is given by
:<math>H = -\sum J_{jk} S_j S_k - \sum z_j S_j</math>
where ''S''<sub>''j''</sub>'s are spin variables, ''z<sub>j</sub>'' external field. 
The system is said to be '''ferromagnetic''' if all the coefficients in the interaction term ''J''<sub>''jk''</sub> are non-negative reals.

The [[Physics:Partition function (statistical mechanics)|partition function]] is given by
:<math>Z = \int e^{- H} d\mu_1(S_1)\cdots d\mu_N(S_N)</math>
where each ''dμ''<sub>''j''</sub> is an even [[Measure (mathematics)|measure]] on the reals '''R''' decreasing at infinity so fast that all [[Gaussian function]]s are integrable, i.e.
:<math> \int e^{b S^2} d|\mu_j(S)| < \infty , \, \forall b \in \mathbb{R}.</math>

A rapidly decreasing measure on the reals is said to have the '''Lee-Yang property''' if all zeros of its Fourier transform are real as the following.
:<math> \int e^{h S} d\mu_j(S) \neq 0 , \, \forall h \in \mathbb{H}_{+} := \{ z \in \mathbb{C} \mid \Re(z)>0 \} </math>

===Theorem===
The '''Lee–Yang theorem states''' that if the Hamiltonian is ferromagnetic and all the measures ''dμ''<sub>''j''</sub> have the Lee-Yang property, and all the numbers ''z''<sub>''j''</sub> have positive real part, then 
the partition function is non-zero. 	
:<math> Z(\{ z_j \}) \neq 0 , \, \forall z_j \in \mathbb{H}_{+}</math>
In particular if all the numbers ''z''<sub>''j''</sub> are equal to some number ''z'', then all zeros of the partition function (considered as a function of ''z'') are imaginary.

In the original Ising model case considered by Lee and Yang, the measures all have support on the 2 point set −1, 1, 
so the partition function can be considered a function of the variable ρ = ''e''<sup>π''z''</sup>. With this change of variable the Lee–Yang theorem says that all zeros ρ lie on the unit circle.

==Examples==

Some examples of measure with the Lee–Yang property are:
*The measure of the Ising model, which has support consisting of two points (usually 1 and &minus;1) each with weight 1/2. This is the original case considered by Lee and Yang.
*The distribution of spin ''n''/2, whose support has ''n''+1 equally spaced points, each of weight 1/(''n''&nbsp;+&nbsp;1). This is a generalization of the Ising model case.
*The density of measure uniformly distributed between &minus;1 and 1.
*The density <math>\exp(-\lambda\cosh(S))\,dS</math>
*The density <math>\exp(-\lambda S^4-bS^2)\,dS</math> for positive λ and real ''b''. This corresponds to the (''&phi;''<sup>4</sup>)<sub>2</sub> Euclidean quantum field theory.
*The density <math>\exp(-\lambda S^6- aS^4-bS^2)\,dS</math> for positive λ does not always have the Lee-Yang property.
*If dμ has the Lee-Yang property, so does exp(''bS''<sup>2</sup>)&nbsp;''d&mu;'' for any positive ''b''.
*If ''d&mu;'' has the Lee-Yang property, so does ''Q''(''S'')&nbsp;''d&mu;'' for any even polynomial ''Q'' all of whose zeros are imaginary.
*The convolution of two measures with the Lee-Yang property also has the Lee-Yang property.

== See also ==
*[[Physics:Lee–Yang theory|Lee–Yang theory]]

==References==
*{{Citation | last1=Itzykson | first1=Claude | last2=Drouffe | first2=Jean-Michel | title=Statistical field theory. Vol. 1 | publisher={{wipe|Cambridge University Press}} | series=Cambridge Monographs on Mathematical Physics | isbn=978-0-521-34058-8 |mr=1175176 | year=1989}}
*{{Citation | last1=Knauf | first1=Andreas | title=Number theory, dynamical systems and statistical mechanics |mr=1714352 | year=1999 | journal=Reviews in Mathematical Physics | issn=0129-055X | volume=11 | issue=8 | pages=1027–1060|doi=10.1142/S0129055X99000325| bibcode=1999RvMaP..11.1027K | citeseerx=10.1.1.184.8685 }}
*{{Citation | last1=Lee | first1=T. D. | last2=Yang | first2=C. N. | title=Statistical Theory of Equations of State and Phase Transitions. II. Lattice Gas and Ising Model | doi=10.1103/PhysRev.87.410  | year=1952 | journal=[[Physics:Physical Review|Physical Review]] | issn=0031-9007 | volume=87 | issue=3 | pages=410–419| bibcode=1952PhRv...87..410L }}
*{{Citation | doi=10.1007/BF01213009 | last1=Lieb | first1=Elliott H. | last2=Sokal | first2=Alan D. | title=A general Lee-Yang theorem for one-component and multicomponent ferromagnets |mr=623156 | year=1981 | journal=Communications in Mathematical Physics | issn=0010-3616 | volume=80 | issue=2 | pages=153–179| bibcode=1981CMaPh..80..153L | s2cid=59332042 |url=http://projecteuclid.org/euclid.cmp/1103919874}}
*{{Citation | last1=Newman | first1=Charles M. | title=Zeros of the partition function for generalized Ising systems | doi=10.1002/cpa.3160270203 |mr=0484184 | year=1974 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=27 | issue=2 | pages=143–159}}
*{{Citation | doi=10.1007/BF01645626 | last1=Simon | first1=Barry |  last2=Griffiths | first2=Robert B. | title=The (φ<sup>4</sup>)<sub>2</sub> field theory as a classical Ising model | url=http://projecteuclid.org/euclid.cmp/1103859251 |mr=0428998 | year=1973 | journal=Communications in Mathematical Physics | issn=0010-3616 | volume=33 | issue=2 | pages=145–164| bibcode=1973CMaPh..33..145S | citeseerx=10.1.1.210.9639 | s2cid=123201243 }}
*{{Citation | last1=Yang | first1=C. N. | last2=Lee | first2=T. D. | title=Statistical Theory of Equations of State and Phase Transitions. I. Theory of Condensation | doi=10.1103/PhysRev.87.404 | year=1952 | journal=[[Physics:Physical Review|Physical Review]] | issn=0031-9007 | volume=87 | issue=3 | pages=404–409| bibcode=1952PhRv...87..404Y }}

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